Transcript Paper for ITPA

SWIP
托卡马克位形优化 (2)
高庆弟
核工业西南物理研究院 成都
1
Nonlinearity of LH wave absorption
The plasma temperature in HL-2A is much lower than that in future
reactor. To establish RS configuration, the LH driven current should be
located off-axis where the plasma temperature is even lower, and the
plasma absorption of high phase velocity LH waves is too weak to ensure
the waves are damped during their first pass.
In the weak electron Landau damping condition LH wave rays make many passes
through the wave propagation domain in plasma and undergo numerous reflections
at the propagation boundaries. Considering the propagation of lower-hybrid waves in
a tokamak. The phase space energy density of the rf field is denoted by U( x, k, t),
where x is the position vector, k is the wave vector. U obeys the WKE,
After determining U( x, k), we can calculate physical quantities such as the (timeaveraged) absorbed power density,
2
The (time-averaged) energy density in the rf parallel electric field,
Here //  d /  // (d /  ) follows from the cold plasma dispersion relation. The
damping rate e for electron Landau damping can be written as follows:
THE CYLINDRICAL APPROXIMATION
Appropriate canonical coordinates in tokamak geometry are (x, k)(r, , , kr, m, n).
We consider the source S = Pin(2)-2 (r - ro) (kr – kr0), the solution of ( 1) is
where
r 2 dr 2 ( r )
dr ( r)
 
y( r )   
,
with
r1
r1 v ( r )
vr ( r )
r
1
r
The absorbed power density,
3
The above solution of the WKE
can be classified into two
distinct para-meter regimes: (i)
the multipass regime (for   1),
when U tends to be uniform
along the entire ray orbit in the
(r, k) plane, and (ii) the singlepass regime (for  > 5) when
nearly all the power is absorbed
before the ray reaches the
caustic. In the multi-pass
regime, the absorption due to
electron Landau damping is
strongly peaked at the caustic.
Fig. 10 Radial profiles of P and |E//|2
for multipass absorption of a single
field harmonic (m = 100, n = 450) in
the cylindrical approximation. 4
In the tokamak toroidal geometry
intrinsic poloidal asymmetry breaks
the invariance of m, causing
formation of a thick stochastic layer
in the ray phase space.
In the LHCD discharges on Tore
Supra, a regime with stationary
oscillation behavior has been
observed because of the nonlinearly
coupling effect of both wave-plasma
interaction and turbulence
suppression by the RS q profile. It is
interpreted as that the current density
and electron temperature profiles
behave as a predator-prey system
[Giruzzi, G., et al., Phys. Rev. Lett. 91
Fig, 11 Surface-s of section in the (m, )
plane for two parameter sets on Tore Supra. (2003) 135001].
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 LH wave absorption in a quasi-stationary RS plasma
I.
Simplified dispersion relation
When the WKB approximation is valid, the wave
matrix
equation is

  2



2
[kk  I k  k 0 K (r , k ,  )]  E  0
(1)

wher k 0   / c , K is the dielectric tensor. For a non-
trivial solution,

 
  2


2
D( , k , r )  [k k  I k  k0 K (r , k ,  )]  0
(2)
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 ci     ce
For LH waves,
K xx  K yy

    k ,  K yx  K xy  i xy  i  ,
ce
2
pe
K zz   //  iK zz, i
where


  1 


j 
2
pe
2
ce
2
pi , j
2
,

 //  1   /  , K zz,i
2
pe
2
3
2
pe
4
4
ce
v  3
2
Te

 

2
pe
j

2
2
pi, j Ti , j
v

4
f e
 dv// v// v//  (  k // v// )
7
If n  kc /   1, a simplified dispersion relation can be found
from the matrix equation by asymptotically expanding
  0 1 1
 1 
n ~ n  2 n  o 4 
n
 n ,
 0
1 1
 1 
E ~ E  2 E  o 4 
n
n 
With the assumption
0
E
 0
1
E  K E  2 ,
n
 0
to the lowest order, I   E  0 ,
 0  0 0
E n E
(3)
where

  
I   I  n n , so that
8
In first order, there arises the solubility condition:
0  0
n K n  0
This gives the simplified dispersion relation
(electrostatic limit)
Dr  k   k   k  0
4

2
 
2
// //
For cold plasmas (  0) ,
k  k ( /  ) /  
2

2
//
2
pe
2
The consistency condition (3) is satisfied for n   / 
2
//
2
pe
2
ce
(~0.5-0.12)
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LH wave absorption regime
 Strong Landau Damping Limit
If the LH wave phase velocity is higher than 3.5 times the
electron thermal velocity, there are too few velocity-resonant
electrons to carry driven current density comparable with the
ohmic current density.
k // c
6.5
n// 


Te [kev]
 n// - Upshift Boundary
In the simulated quasi-stationary RS discharges, it turns out
19
3
Te01.4kev (Ti02.8kev) with n e  2.32  10 m . In such
conditions there is a spectral gap between the parallel LHW
phase velocity and the electron thermal velocity.
10
B
B
2
2
n//  n 
n  nr
,
B
B
where n is the wave vector component perpendicular to
the magnetic field. We are interested in the maximum
upshift factor of n// , taking nr  0 , which applies at a
radial turning point. In tokamak plasmas,
n//  n  n B / B .
By using the cold electrostatic approximation of the
dispersion relation,
R0 / R
n//  n// 0
1  ( pe /  ) /( qˆ   )
q cyl
ˆ
q
 a/R .
with
,
where
x
11
 Propagation Domain
At the boundary of the propagation domain,
n  n  n  n
2

2
//
2
2
(7)
As the tokamak equilibrium is toroidal axisymetric, the toroidal mode
number n is conserved. From the definition of n// ,
n//
n x
1
n
n qcyl ( x)
(8)
From Eq. (8),
2
2
n//
n// 2
n
2
ˆ

1


q
(1  )
2
2
n
n
n
(9)
By using the cold electrostatic approximation of the dispersion relation,
R0 qˆ  1  (1  qˆ )( /  ) /  
2
2
2
ˆ
ˆ
R
q  [1  ( pe /  ) /   ]
2
n//  n// 0
with
2
2
pe
2


  1 



j
2
pe
2
ce
2
pi , j
2
12
Fig. 1 Evolution of LH wave driven
current profile for the case of the
LH spectrum produced with in (a)
the Ip = 265kA discharge, and (b)
the Ip = 300KA discharge
Fig. 2 Region of LH power absorption
(at t=1.0s): electron Landau damping
limit (full line); n// -upshift boundary
(dotted line); and boundary of the
wave propagation domain ( dash line).
(a) Ip = 265KA, (b) Ip = 300KA
13
The LH wave absorption is bounded in the region defined by the strong Landaudamping limit and the boundary of wave propagation domain. This mechanism of
the LH wave absorption causes interplay of the distribution of the LH wave driven
current with the modification of the plasma configuration, which constitutes nonlinearity in the LH wave deposition.
Due to nonlinearity of the
LH power
absorption, the
LH wave
deposition
position changes
spontaneously,
generating two
distinct quasistationary
reversed
magnetic shear
(RS) configurations.
Fig.3 Time evolution of the LH wave driven current profile, and
the regions of LH power absorption by strong electron Landau
damping at two different times: (a) t=1.5s, (b) t=0.8s
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non-predator-prey oscillation
In a NBI heated plasma of
Ip = 265kA, BT = 2.8T, and
, ne  2.32  1019 m 3 by
controlling the radiated LH
spectrum (PLH = 0.5MW),
quasi-stationary RS discharge
[Q. Gao, et al. Nucl Fusion
43 (2003) 982], two-phase
RS discharge [Q. Gao, et al.
Phys Plasmas 12 (2005)
122507] have been obtained.
When  = 60 the location
of the peak of LH driven
current presents oscillation
with irregular cyclic.
Fig. 4 Time evolution of (a) location of peak of
the LH driven current profile, and (b) location of
the minimum q in the oscillating RS discharge
(full line), the two-phase RS discharge, and the
stationary RS discharge (dashed line)
15
(a) Profiles of the ion
temperature at t=1.42s
(dotted line), and t=1.69s
(full line) in the
oscillating RS discharge.
(b) Ion thermal diffusivity
i versus  at t=1.42s
(dotted line), and t=1.69s
(full line) with the
corresponding thin lines
indicating neo-classical
value.
16
Oscillations during LH ramp-up in Tore Supra
17
Predator-prey systems
• Lotka-Volterra equations
• Used for modelling populations
in ecosystems
• 2 coupled nonlinear equations,
periodic solutions
• J = predator; T = prey ?
 

JT 
t J   J J 
1
T    J J   JT JT

 J 

 
TJ

t T   T T 1
T


J 
   T T  TJ TJ

18
Resistive diffusion and heat transport equations:
J 1 
0

r (T )( J  JLH ( j ,T )  J bs( j ,T ))
 t r r r
T
1  

T 
n
 PLH ( J,T ) 
r n ( J,T )  losses coupling with ions
t
r r 
r 
 ( J,T )  c1  c2H ( s), s  shear ,  resistivity
...
have similarities with the Lotka-Volterra equations:
1 
r (J  J LH  J bs )    J J(1 bT)
r r r
1  
T 
PLH (J,T) 
rn (J,T)    T T(1 aJ)
r r 
r 
dJ
dT
  J J(1 bT)
  T T(1 aJ)
dt
dt
19
Oscillations reproduced by CRONOS
coupled resistive and heat diffusion equations
have periodic solutions if:
• jLH(r)  j(r)Te(r)
• e is a function of j (e.g., improved confinement for negative shear)
Outputs of the
CRONOS code
• resistive diffusion
• heat transport
• self-cons. equilibrium
• jLH(r)  j(r)Te(r)
20
The oscillatory behavior in the LHCD controlled discharge on
HL-2A is induced by nonlinear coupling of the LH power
absorption position with the plasma configuration.
According to the analysis by using the wave kinetic equation, in
the weak damping regime absorption of the LH waves due to ELD
is strongly peaked at caustic.
The peak location of LH driven current is determined by the
intersection between the inner boundary (caustic) of the propagation
domain and the ELD limit.
21
The mechanism of LH power deposition region coupling the q profile
and Te profile is the following:
In the tokamak plasma condition, the upper boundary of the LH wave
propagation domain is reduced to 
q
n// u  (n// 0 R0 / R )  
q 
In the central plasma region ( < 0.7), is nearly a constant
approximately equivalent to 18 because it is mainly dependent on
square-root of the plasma density. Thus is a decreasing function of
the magnetic geometry factor;
 The ELD limit is a decreasing function of Te, and actually it is nearly
unchanged in the oscillation since it is inversely proportional to
square root of Te.
Whenever the LH driven current moves inwards, decreases due to
the safety factor decreasing, the caustic boundary is elevated. The
intersection between the caustic boundary and the ELD limit would
22
move inward further
Variation of the LH deposition region defined by the wave propagation condition and strong Landau damping shows consistency with the plasma oscillation.
In the vicinity of min the oscillating amplitude of the inverse pitch angle of
magnetic field (BT / Bp) is quite large. Therefore, it is feasible to measure the
oscillation with MSE in experiments.
Fig. 5 LH power absorption region in the
phasing space (, n//) defined by ELD limit
and inner boundary of the wave propagation
domain at t=1.42s (dotted line), and t=1.65s
(full line) in the oscillating RS discharge.
Fig. 6 Oscillation of the inverse pitch
angle of the magnetic field (BT / Bp).
23